On Sat, 17 Jul 2004, Donald Stahl wrote:
> There is learnability and learnability. In one sense, plenty of people have
> learned English. In another, it is plausible that no one has ever learned
> English, in that it is plausible that no one has ever learned all the words
> in the OED. When Davidson introduced the term 'learnability' into
> philosophical discussion he had it tied very closely to the finite.
Davidson tied learnability (of a language) to finitude of the *lexicon*;
then he stressed the potential infinity of expressions that could be
understood by a finite being, and concluded that there must be some
finitely stateable, effective method for assigning interpretations to
expressions. Given the potential infinity of interpretable expressions,
the method in question (according to Davidson) would be inductive or
recursive. And that led him to stress the Tarskian method of inductively
generating truth-conditions for sentences of an object language.
The lexicon of extra-logical expressions in the language of first-order
arithmetic is finite. Indeed, there are only four such expressions: 0, s,
+ and x . Then there are (say) four connectives, two quantifiers, and
the identity predicate. The need for infinitely many variables (for the
formation of arbitrarily long but finite quantified sentences) can be
ignored here; a similar problem would arise for natural languages, with
anaphoric pronouns within sentences of increasing length. Formal grammar
provides an effective method for determining, of any finite string of
symbols, whether it is a well-formed sentence in the language of Peano
arithmetic.
So: the *language* of Peano arithmetic is learnable, in Davidson's sense.
But what about the *theory*? Here, we need to distinguish between the set
of all first-order truths in the standard model, and the set of theorems
provable from the Peano axioms. I claim only that the latter theory is
learnable, in any reasonable sense of "learnable". A theory can be
learnable without theoremhood in that set being effectively decidable. All
that has to be learned is (i) a method for recognizing, of any given
sentence of the language, whether it is an axiom of the theory; and (ii)
the permissible forms of primitive steps of inference by means of which
theorems can be proved from axioms. Together, (i) and (ii) ensure that we
have (iii) an effective method for determining, of any given finite
piece of discourse P, and any given sentence S, whether P is a proof of S
from the axioms of Peano arithmetic.
As finite beings we can obviously acquire both (i) and (ii), hence (iii).
And this is the case despite the fact that the first-order theory of Peano
arithmetic is not finitely axiomatizable.
Note that every axiomatized, but not finitely axiomatizable, theory in
mathematics has been axiomatized by using at most *finitely many* axiom
schemes.
Neil Tennant